# NAG FL Interfacef06wnf (zlanhf)

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## 1Purpose

f06wnf returns the value of the $1$-norm, the $\infty$-norm, the Frobenius norm, or the maximum absolute value of the elements of a complex Hermitian matrix $A$ stored in Rectangular Full Packed (RFP) format.

## 2Specification

Fortran Interface
 Function f06wnf ( norm, uplo, n, a, work)
 Real (Kind=nag_wp) :: f06wnf Integer, Intent (In) :: n Real (Kind=nag_wp), Intent (Inout) :: work(*) Complex (Kind=nag_wp), Intent (In) :: a(n*(n+1)/2) Character (1), Intent (In) :: norm, transr, uplo
#include <nag.h>
 double f06wnf_ (const char *norm, const char *transr, const char *uplo, const Integer *n, const Complex a[], double work[], const Charlen length_norm, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by the names f06wnf, nagf_blas_zlanhf or its LAPACK name zlanhf.

## 3Description

Given a complex $n×n$ symmetric matrix, $A$, f06wnf calculates one of the values given by
 ${‖A‖}_{1}=\underset{j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{i=1}^{n}|{a}_{ij}|$ (the $1$-norm of $A$), ${‖A‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{j=1}^{n}|{a}_{ij}|$ (the $\infty$-norm of $A$), ${‖A‖}_{F}={\left(\sum _{i=1}^{n}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}\right)}^{1/2}$ (the Frobenius norm of $A$),   or $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ (the maximum absolute element value of $A$).
$A$ is stored in compact form using the RFP format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

## 4References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee https://www.netlib.org/blas/blast-forum/blas-report.pdf
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## 5Arguments

1: $\mathbf{norm}$Character(1) Input
On entry: specifies the value to be returned.
${\mathbf{norm}}=\text{'1'}$ or $\text{'O'}$
The $1$-norm.
${\mathbf{norm}}=\text{'I'}$
The $\infty$-norm.
${\mathbf{norm}}=\text{'F'}$ or $\text{'E'}$
The Frobenius (or Euclidean) norm.
${\mathbf{norm}}=\text{'M'}$
The value $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ (not a norm).
Constraint: ${\mathbf{norm}}=\text{'1'}$, $\text{'O'}$, $\text{'I'}$, $\text{'F'}$, $\text{'E'}$ or $\text{'M'}$.
2: $\mathbf{transr}$Character(1) Input
On entry: specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'C'}$.
3: $\mathbf{uplo}$Character(1) Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
When ${\mathbf{n}}=0$, f06wnf returns zero.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{a}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$Complex (Kind=nag_wp) array Input
On entry: the upper or lower triangular part (as specified by uplo) of the $n×n$ Hermitian matrix $A$, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the F07 Chapter Introduction.
6: $\mathbf{work}\left(*\right)$Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{norm}}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$, and at least $1$ otherwise.

None.

## 7Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

## 8Parallelism and Performance

f06wnf is not threaded in any implementation.

None.

## 10Example

This example reads in the lower triangular part of a symmetric matrix, converts this to RFP format, then calculates the norm of the matrix for each of the available norm types.

### 10.1Program Text

Program Text (f06wnfe.f90)

### 10.2Program Data

Program Data (f06wnfe.d)

### 10.3Program Results

Program Results (f06wnfe.r)